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Discussiones Mathematicae Graph Theory 28 (2008 ) 59–66 TREES WITH EQUAL TOTAL DOMINATION AND TOTAL RESTRAINED DOMINATION NUMBERS Xue-Gang Chen∗ Department of Mathematics North China Electric Power University Beijing 102206, China e-mail: gxc [email protected] Wai Chee Shiu Department of Mathematics Hong Kong Baptist University 224 Waterloo Road, Kowloon Tong, Hong Kong, China and Hong-Yu Chen The College of Information Science and Engineering Shandong University of Science and Technology Qingdao, Shandong Province 266510, China Abstract For a graph G = (V, E), a set S ⊆ V (G) is a total dominating set if it is dominating and both hSi has no isolated vertices. The cardinality of a minimum total dominating set in G is the total domination number. A set S ⊆ V (G) is a total restrained dominating set if it is total dominating and hV (G) − Si has no isolated vertices. The cardinality of a minimum total restrained dominating set in G is the total restrained domination number. We characterize all trees for which total domination and total restrained domination numbers are the same. Keywords: total domination number, total restrained domination number, tree. 2000 Mathematics Subject Classification: 05C69. ∗ Partially Supported by CERG Research Grant Council of Hong Kong and Faculty Research Grant of Hong Kong Baptist University. Supported by Doctoral Research Grant of North China Electric Power University (200722026). 60 X.-G. Chen, W.C. Shiu and H.-Y. Chen 1. Introduction By a graph we mean a finite, undirected graph without loops or multiple edges. Terms not defined here are used in the sense of Arumugam [1]. Let G = (V, E) be a simple graph of order n. The degree, neighborhood and closed neighborhood of a vertex v in the graph G are denoted by d G (v), NG (v) and NG [v] = NG (v)∪{v}, respectively. For a subset S of V , N G (S) = S v∈S NG (v) and NG [S] = NG (S) ∪ S. The graph induced by S ⊆ V is denoted by hSi. The minimum degree and maximum degree of the graph G are denoted by δ(G) and ∆(G), respectively. The diameter diam(G) of a connected graph G is the maximum distance between two vertices of G, that is diam(G) = maxu,v∈V (G) dG (u, v). Let Pn denote a path with n vertices. Let K1,r denote the star with r +1 vertices. Define K 1,r,4 as follows: for each edge of K1,r , we subdivide by two vertices. The vertex of degree r is called the central vertex of K1,r,4 . Let η be a family of graphs and η = {K1,r,4 |r ≥ 1 and r is an integer }. A subset S of V is called a dominating set if every vertex in V − S is adjacent to some vertex in S. The domination number γ(G) of G is the minimum cardinality taken over all dominating sets of G. A set S ⊆ V (G) is a total dominating set if it is dominating and hSi has no isolated vertices. The cardinality of a minimum total dominating set in G is the total domination number and is denoted by γ t (G). Cockayne et al. [6] studied total dominating functions in trees: minimality and convexity. The total restrained domination number of a graph was defined by D. Ma et al. in [4]. A set S ⊆ V (G) is a total restrained dominating set if it is total dominating and hV (G) − Si has no isolated vertices. The cardinality of a minimum total restrained dominating set in G is the total restrained domination number and is denoted by γ rt (G). A total dominating set S with cardinality γ t (G) is called a γt -set. A total restrained dominating set S with cardinality γ rt is called a γrt -set. Let S ⊂ V (G) and x ∈ S, we say that x has a private neighbour (with respect to S) if there is a vertex in V (G) − S whose only neighbour in S is x. Let P N (x, S) denote the private neighbours set of x with respect to S. A vertex of degree one is called a leaf. A vertex v of G is called a support if it is adjacent to a leaf. If T is a tree, L(T ) and S(T ) denote the set of leaves and supports, respectively. Any vertex of degree greater than one is called an internal vertex. Trees with Equal Total Domination and ... 61 For any graph theoretical parameters λ and µ, we define G to be (λ, µ)-graph if λ(G) = µ(G). In this paper we provide a constructive characterization of (γt , γrt )-trees. 2. A Characterization of (γt , γrt )-trees As a consequence of the definition of total restrained domination number, we have the following observations. Observation 1. Let G be a graph without isolated vertices. Then (i) every leaf belongs to every γrt -set; (ii) every support belongs to every γ rt -set; (iii) γt (G) ≤ γrt (G). Observation 2. Let T be a (γt , γrt )-tree. Then each γrt (T )-set is a γt (T )-set. Let τ1 and τ2 be the following two operations defined on a tree T . • Operation τ1 . Assume x ∈ V (T ) is a leaf or support. Then add one or more trees of η and the edges between x and each central vertex. • Operation τ2 . Assume x ∈ N (S(T )) − L(T ). Then add one or more paths P3 and the edges between x and one leaf of each P 3 . Let τ be the family of trees such that τ = {T : T is obtained from P 6 by a finite sequence of operations τ1 or τ2 } ∪ {P2 , P6 }. We show first that each tree in the family τ has equal total domination number and total restrained domination number. Lemma 1. If T belongs to the family τ , then T is a (γ t , γrt )-tree. P roof. We proceed by induction on the number of operations s(T ) required to construct the tree T . If s(T ) = 0, then T ∈ {P 2 , P6 } and clearly T is a (γt , γrt )-tree. Assume now that T is a tree with s(T ) = k for some positive integer k and each tree T 0 ∈ τ with s(T 0 ) < k is a (γt , γrt )-tree. Then T can be obtained from a tree T 0 belonging to τ by operation τ1 or τ2 . We now consider two possibilities depending on whether T is obtained from T 0 by operation τ1 or τ2 . 62 X.-G. Chen, W.C. Shiu and H.-Y. Chen Case 1. T is obtained from T 0 by operation τ1 . Without loss of generality, we can assume that T is obtained from T 0 by adding k trees K1,r1 ,4 , K1,r2 ,4 , . . . , K1,rk ,4 of η and the edges between x and each central vertex, where r1 ≤ r2 ≤ · · · ≤ rk . It is obvious that γt (T ) ≤ γt (T 0 ) + P 2 1≤i≤k ri . Let D be a γt -set of T such that D ∩ L(T ) = ∅. Then |D ∩ K1,ri ,4 | ≥ 2ri for each K1,ri ,4 . Let D 0 = D ∩ V (T 0 ). Case 1.1. x is a support of T 0 . Then x ∈ D 0 . If NT 0 (x) ∩ P D 0 6= ∅, then D 0 is a total dominating set of T 0 . So γt (T 0 ) ≤ |D 0 | ≤ γt (T ) − 2 1≤i≤k ri . If NT 0 (x)∩D 0 = ∅, then there exists a tree K1,ri ,4 such that |D∩K1,ri ,4 | ≥ 2ri +1 and its central vertex belongs to D. Let y ∈ N T 0 (x) and D 00 = D 0 ∪ {y}. Then D 00 P is a total dominating set of T 0 . So γt (T 0 ) ≤ |D 00 | = |D 0 | + 1 ≤ γt (T ) − 2 1≤i≤k ri . Case 1.2. x is a leaf of T 0 . Let y ∈ NT 0 (x). If y ∈ D, then D 0 is a total dominating set of T 0 . Suppose y ∈ / D. Then there exists a tree K1,ri ,4 such that |D ∩ K1,ri ,4 | ≥ 2ri + 1 and its central vertex belongs to D. Let D 00 = D 0 ∪ {y}. Then D 00Pis a total dominating set of T 0 . So γt (T 0 ) ≤ |D 00 | = |D 0 | + 1 ≤ γt (T ) − 2 1≤i≤k ri . P Hence, γt (T ) = By Case 1.1 and 1.2, γt (T 0 ) ≤ γt (T ) − 2 1≤i≤k ri . P P t (T ) ≤ γ t (T 0 ) + 2 that γ γt (T 0 ) + 2P 1≤i≤k ri . It is obvious r r 1≤i≤k ri . Since P t t 0 0 γr (T )+2 P1≤i≤k ri = γt (T )+2 1≤i≤k ri = γt (T ) ≤ γr (T ). Hence γrt (T ) = γrt (T 0 ) + 2 1≤i≤k ri . So γt (T ) = γrt (T ). Case 2. T is obtained from T 0 by operation τ2 . Without loss of generality, we can assume that T is obtained from T 0 by adding paths v1j , v2j , v3j and the edges between x and v1j for j = 1, 2, · · · , k. It is obvious that γt (T ) ≤ γt (T 0 ) + 2k. Let D be a γt -set of T such that D ∩ L(T ) = ∅. Then v1j , v2j ∈ D. Let D 0 = D ∩ V (T 0 ). Then D 0 is a total dominating set of T 0 . So γt (T 0 ) ≤ γt (T ) − 2k. Hence γt (T ) = γt (T 0 ) + 2k. Let D 00 be a γrt -set of T 0 . Since T 0 is a (γt , γrt )-tree, it follows that x ∈ / D 00 . Otherwise, assume NT 0 (x) ∩ S(T 0 ) = {y} and NT 0 (y) ∩ L(T 0 ) = {z}. Then D 00 − {z} is a total dominating set of T 0 with cardinality less than |D 00 |, which is a contradiction. So, γrt (T ) ≤ γrt (T 0 ) + 2k. Since γrt (T 0 ) + 2k = γt (T 0 ) + 2k = γt (T ) ≤ γrt (T ). Hence γrt (T ) = γrt (T 0 ) + 2k. So γt (T ) = γrt (T ). We show next that every (γt , γrt )-tree belongs to the family τ . Trees with Equal Total Domination and ... 63 Lemma 2. Let T be a (γt , γrt )-tree. Then (i) for each support v ∈ S(T ), |N (v) ∩ L(T )| = 1; (ii) for any two supports u, v ∈ S(T ), d(u, v) ≥ 3. P roof. (i) Suppose that there exists a support v such that |N (v) ∩ L(T )| ≥ 2. Let N (v) ∩ L(T ) = {v1 , . . . , vk } where k ≥ 2. Let D be a γrt -set of T . Then, by Observation 1, it follows that D−{v 2 , . . . , vk } is a total dominating set of T with cardinality less than γ t (T ), which is a contradiction. Hence, |N (v) ∩ L(T )| = 1 for each support v ∈ S(T ). (ii) Suppose that there exist two supports u and v such that d(u, v) ≤ 2. Let u1 ∈ N (u) ∩ L(T ) and v1 ∈ N (v) ∩ L(T ). Let D be a γrt -set of T . If u is adjacent to v, then, by Observation 1, it follows that D − {u 1 } is a total dominating set of T with cardinality less than γ t (T ), which is a contradiction. Suppose d(u, v) = 2. Assume w ∈ N (u) ∩ N (v). Then by Observation 1, it follows that (D − {u 1 , v1 }) ∪ {w} is a total dominating set of T with cardinality less than γ t (T ), which is a contradiction. Hence, d(u, v) ≥ 3 for any two supports u, v ∈ S(T ). Lemma 3. If T is a (γt , γrt )-tree, then T belongs to the family τ . P roof. Let T be a (γt , γrt )-tree. If diam(T ) ≤ 5, then T is P2 or P6 . It is clear that the statement is true. For this reason, we only consider only trees T with diam(T ) ≥ 6. Let T be a (γt , γrt )-tree and assume that the result holds for all trees on n(T ) − 1 and fewer vertices. We proceed by induction on the number of vertices of a (γt , γrt )-tree. Let P = (v0 , v1 , . . . , vl ), l ≥ 6, be a longest path in T and let D be a γrt (T )-set. Then v0 , v1 ∈ D. By Lemma 2, it follows that d(v1 ) = d(v2 ) = 2. It is obvious that v2 , v3 ∈ / D. Otherwise D − {v0 } is a total dominating set with cardinality less than |D|, which is a contradiction. Now we have the following claim. Claim 1. |NT (v3 ) ∩ D| = 1. P roof. Without loss of generality, we can assume |N T (v3 ) ∩ D| = t and t > 1. Then NT (v3 )∩D ⊆ S(T )∪{v4 }. By Lemma 2, |NT (v3 )∩D∩S(T )| = 1. So, t = 2. We can assume NT (v3 ) ∩ D = {v31 , v4 }, where v31 ∈ S(T ). By Lemma 2, it is easy to prove that v5 ∈ D. Let A1 = NT (v5 ) − {v4 }. 64 X.-G. Chen, W.C. Shiu and H.-Y. Chen Then for any v ∈ A1 , v ∈ / D. Otherwise, let T1 denote the component of T − {v5 } containing v4 . Then (D − (L(T1 ) ∪ {v4 })) ∪ (NT1 [S(T1 )] − L(T1 )) is a total dominating set of T with cardinality less than |D|, which is a contradiction. Let B1 = NT (A1 ) ∩ (V (T ) − D), A2 = NT (B1 ) ∩ D and B2 = NT (A2 ) ∩ D. For i ≥ 1, let A2i+1 = NT (B2i ) ∩ (V (T ) − D), B2i+1 = NT (A2i+1 )∩(V (T )−D), A2i+2 = NT (B2i+1 )∩D and B2i+2 = NT (A2i+2 )∩D. It is obvious that |B2i+1 | ≤ |A2i+2 | ≤ |B2i+2 | for i ≥ 0. Now we prove that if NT (B2i+2 ) ∩ D − A2i+2 6= ∅, then |NT (v) ∩ D| ≥ 2 for any v ∈ NT (B2i+2 ) ∩ D − A2i+2 . Otherwise, we can assume t is the maximum i satisfying NT (B2i+2 ) ∩ D − A2i+2 6= ∅ and there exists a vertex v ∈ NT (B2i+2 ) ∩ D − A2i+2 such that |NT (v) ∩ D| = 1. Without loss of generality, we can assume that u ∈ B2t+2 and uv ∈ E(T ). Define C1 = NT (v) \ {u}. Then for any w ∈ C1 , w ∈ / D. Let D1 = NT (C1 ) ∩ (V (T ) − D). Let C2 = NT (D1 ) ∩ D and D2 = NT (C2 ) ∩ D. For i ≥ 1, let C2i+1 = NT (D2i ) ∩ (V (T ) − D), D2i+1 = NT (C2i+1 ) ∩ (V (T ) − D), C2i+2 = NT (D2i+1 ) ∩ D and D2i+2 = NT (C2i+2 ) ∩ D. It S is obvious that 0 |D2i+1 | ≤ |C2i+2 | ≤ |D2i+2 | for i ≥ 0. Let D = (D − {v} − 0≤i≤t D2i+2 ) ∪ S 0 0≤i≤t D2i+1 . It is obvious that D is a total dominating set of T with cardinality less than |D|, which is a contradiction. S Let w ∈ A1 . Let D = (D − (L(T1 ) ∪ {v4 , v5 }) − 0≤i≤t B2i+2 ) ∪ S 0≤i≤t B2i+1 ∪ {w} ∪ (NT1 [S(T1 ))] − L(T1 )). It is obvious that D is a total dominating set of T with cardinality less than |D|, which is a contradiction. Hence, |NT (v3 ) ∩ D| = 1. By the above claim, we consider the following three cases. dT (v4 ) = j. Assume Case 1. v4 ∈ D and v4 ∈ S(T ). Let T1 denote the component of T − {v4 } containing v5 . Let NT (v4 ) ∩ L(T ) = {l} and NT (v4 ) − {v5 , l} = {v41 , · · · , v4(j−2) }. Denote T 0 P= hV (T1 ) ∪ {v4 , l}i. Then it is easy to prove that γt (T ) = γt (T 0 ) + 2 1≤i≤(j−2) (dT (v4i ) − 1). It is obvious that P γrt (T 0 ) ≤ γrt (T ) − 2 1≤i≤(j−2) (dT (v4i ) − 1). Since T is a (γt , γrt )-tree, it P follows that γrt (T ) = γt (T ) = γt (T 0 ) + 2 1≤i≤(j−2) (dT (v4i ) − 1) ≤ γrt (T 0 ) + P P 2 1≤i≤(j−2) (dT (v4i )−1). Hence γrt (T ) = γrt (T 0 )+2 1≤i≤(j−2) (dT (v4i )−1). So γt (T 0 ) = γrt (T 0 ). Consequently, T 0 is a (γt , γrt )-tree and by induction hypothesis, T 0 ∈ τ . As v4 is a support in T 0 , we deduce that T may be obtained from T 0 by operation τ1 . Trees with Equal Total Domination and ... 65 Case 2. v4 ∈ D and v4 ∈ / S(T ). Let T1 denote the component of T −{v4 } containing v5 . Then v5 ∈ D. Let NT (v4 ) − {v5 } = {v41 , · · · , v4(j−1)) }. Denote T 0 = hV (T1 ) ∪ {v4 }i. Then it is obvious that γt (T ) = γt (T 0 ) + P P 2 1≤i≤(j−1) (d(v4i ) − 1). It is obvious that γrt (T 0 ) ≤ γrt (T ) − 2 1≤i≤(j−1) t (d(v4i ) − 1). it follows that γrt (T ) = γt (T ) = P Since T is a (γt , γr )-tree, P 0 t 0 γt (T ) + 2 1≤i≤(j−1) (d(v4i ) − 1) ≤ γr (T ) + 2 1≤i≤(j−1) (d(v4i ) − 1). Hence P γrt (T ) = γrt (T 0 )+2 1≤i≤(j−1) (d(v4i )−1). So γt (T 0 ) = γrt (T 0 ). Consequently, T 0 is a (γt , γrt )-tree and by induction hypothesis, T 0 ∈ τ . As v4 is a leaf in T 0 , we deduce that T may be obtained from T 0 by operation τ1 . Case 3. v4 ∈ / D. Then there exists exactly one vertex x ∈ N T (v3 )∩D and x is a support. Assume NT (x) ∩ L(T ) = {l}. Let T1 denote the component of T − {v3 } containing v4 . Denote T 0 = hV (T1 ) ∪ {v3 , x, l}i. It is obvious that γt (T ) = γt (T 0 ) + 2(dT (v3 ) − 2). It is obvious that x, l ∈ D. Hence γrt (T 0 ) ≤ γrt (T ) − 2(dT (v3 ) − 2). Since T is a (γt , γrt )-tree, it follows that γrt (T ) = γt (T ) = γt (T 0 ) + 2(dT (v3 ) − 2) ≤ γrt (T 0 ) + 2(dT (v3 ) − 2). Hence γrt (T ) = γrt (T 0 ) + 2(dT (v3 ) − 2). So γt (T 0 ) = γrt (T 0 ). Consequently, T 0 is a (γt , γrt )-tree and by induction hypothesis, T 0 ∈ τ . As v3 is a vertex adjacent to a support in T 0 , we deduce that T may be obtained from T 0 by operation τ2 . As an immediate consequence of Lemmas 2 and 3 we have the following characterization of (γt , γrt )-trees. Theorem 3. A tree T is a (γt , γrt )-tree if and only if T belongs to the family τ . References [1] S. Arumugam and J. Paulraj Joseph, On graphs with equal domination and connected domination numbers, Discrete Math. 206 (1999) 45–49. [2] G.S. Domke, J.H. Hattingh, S.T. Hedetniemi, R.C. Laskar and L.R. Marcus, Restrained domination in graphs, Discrete Math. 203 (1999) 61–69. [3] F. Harary and M. Livingston, Characterization of tree with equal domination and independent domination numbers, Congr. Numer. 55 (1986) 121–150. [4] D. Ma, X. Chen and L. Sun, On total restrained domination in graphs, Czechoslovak Math. J. 55 (2005) 165–173. [5] G.S. Domke, J.H. Hattingh, S.T. Hedetniemi and L.R. Markus, Restrained domination in trees, Discrete Math. 211 (2000) 1–9. 66 X.-G. Chen, W.C. Shiu and H.-Y. Chen [6] E.J. Cockayne, C.M. Mynhardt and B. Yu, Total dominating functions in trees: minimality and convexity, J. Graph Theory 19 (1995) 83–92. Received 22 September 2006 Revised 24 January 2007 Accepted 24 January 2007